Biomolecular condensates form by phase separation of biological polymers. The cellular functions of the resulting membraneless organelles are closely linked to their physical properties over a wide range of length- and timescales: From the nanosecond dynamics of individual molecules and their interactions, to the microsecond translational diffusion of molecules in the condensates, to their viscoelastic properties at the mesoscopic scale. However, it has remained unclear how to quantitatively link these properties across scales. Here we address this question by combining single-molecule fluorescence, correlation spectroscopy, microrheology, and large-scale molecular dynamics simulations on different condensates that are formed by complex coacervation and span about two orders of magnitude in viscosity and their dynamics at the molecular scale. Remarkably, we find that the absolute timescale of protein chain dynamics in the dense phases can be quantitatively and accurately related to translational diffusion and condensate viscosities by Rouse theory of polymer solutions including entanglement. The simulations indicate that the observed wide range of dynamics arises from different contact lifetimes between amino acid residues, which in the mean-field description of the polymer model cause differences in the friction acting on the chains. These results suggest that remarkably simple physical principles can relate the mesoscale properties of biomolecular condensates to their dynamics at the nanoscale.